H p boundedness of Calder on-Zygmund operators on product domains - PowerPoint PPT Presentation

H p boundedness of Calder´ on-Zygmund operators on product domains Ming-Yi Lee National Central University, TAIWAN Symposium on Probability and Analysis 2010 Institute of Mathematics, Academia Sinica, Taipei, Taiwan • This talk is based on joint work with Yongsheng Han, Chin-Cheng Lin, and Ying-Chieh Lin. All results presented here appeared in J. Funct. Anal. 258 (2010), 2834 – 2861. H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

Calder´ on-Zygmund operators play a central role in modern harmonic analysis. The simplest and most important examples are the Hilbert transform H � f ( x ) Hf ( x ) := p.v. x − y dy R and the Riesz transforms R j , j = 1 , 2 , · · · , n , � x j − y j R j f ( x ) := p.v. | x − y | n +1 f ( y ) dy . R n H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

Calder´ on-Zygmund operators for 1-parameter A singular integral operator T is a continuous linear operator from D ( R n ) into its dual associated to a kernel K ( x , y ), a continuous function defined on R n × R n \ { x = y } , satisfying 1 | K ( x , y ) | ≤ C | x − y | − n ; 2 | K ( x , y ) − K ( x , y ′ ) | ≤ C | y − y ′ | ε if | y − y ′ | ≤ | x − y | ; | x − y | n + ε 2 3 | K ( x , y ) − K ( x ′ , y ) | ≤ C | x − x ′ | ε if | x − x ′ | ≤ | x − y | . | x − y | n + ε 2 The smallest constant C is denoted by | K | CZ . Moreover, the operator T can be represented by � � � Tf , g � = R n K ( x , y ) f ( y ) g ( x ) dydx R n for all f , g ∈ D ( R n ) with supp ( f ) ∩ supp ( g ) = ∅ . H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

Calder´ on-Zygmund operators for 1-parameter (cont.) If a singular integral operator T is bounded on L 2 ( R n ), then we say that this T is a Calder´ on-Zygmund operator and its norm is defined by � T � CZ = � T � L 2 �→ L 2 + | K | CZ . From Calder´ on-Zygmund operator theory, every Calder´ on-Zygmund operator T is bounded on L p ( R n ), p > 1, and bounded from L ∞ ( R n ) to BMO ( R n ). However, T is not bounded on L p ( R n ) for p ≤ 1. Instead of L p ( R n ), one can consider the boundedness of T on Hardy spaces H p ( R n ) when n n + ε < p ≤ 1. H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

Calder´ on-Zygmund operators for 1-parameter (cont.) More precisely, every Calder´ on-Zygmund operator T is bounded on n + ε < p ≤ 1 if and only if T ∗ 1 = 0. The range of p n H p ( R n ), depends on the cancellation of T and smooth condition of its associated kernel. For convolution operator T , K ( x , y ) = K ( x − y ), we do not need T ∗ 1 = 0 to get H p -boundedness of T . H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

L p boundedness on product domain R n × R m In 1982, R. Fefferman and Stein [Adv. in Math.] investigated the L p ( R n + m ) boundedness of convolution operators. Theorem Suppose that K is integrable on R n × R m and satisfies certain “cancellation” and “size” properties. Then � f ∗ K � L p ( R n + m ) ≤ C p � f � L p ( R n + m ) , 1 < p < ∞ . H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

Journ´ e’s class A singular integral operator T defined on R n × R m is said to be in Journ´ e’s class if � Tf ( x 1 , x 2 ) = R n × R m K ( x 1 , x 2 , y 1 , y 2 ) f ( y 1 , y 2 ) dy 1 dy 2 , where the kernel K ( x 1 , x 2 , y 1 , y 2 ) satisfies the following conditions. For each x 1 , y 1 ∈ R n , K 1 ( x 1 , y 1 ) is a Calder´ on-Zygmund operator on R m with the kernel K 1 ( x 1 , y 1 )( x 2 , y 2 ) = K ( x 1 , x 2 , y 1 , y 2 ). Similarly, K 2 ( x 2 , y 2 )( x 1 , y 1 ) = K ( x 1 , x 2 , y 1 , y 2 ). Moreover, H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

Journ´ e’s class (cont.) 1 T is bounded on L 2 ( R n + m ), 2 � K 1 ( x 1 , y 1 ) � CZ ≤ C | x 1 − y 1 | n , 1 ) � CZ ≤ C | y 1 − y ′ 1 | ε 1 |≤ | x 1 − y 1 | � K 1 ( x 1 , y 1 ) − K 1 ( x 1 , y ′ if | y 1 − y ′ , | x 1 − y 1 | n + ε 2 1 , y 1 ) � CZ ≤ C | x 1 − x ′ 1 | ε 1 |≤ | x 1 − y 1 | � K 1 ( x 1 , y 1 ) − K 1 ( x ′ if | x 1 − x ′ , | x 1 − y 1 | n + ε 2 3 � K 2 ( x 2 , y 2 ) � CZ ≤ C | x 2 − y 2 | m , 2 ) � CZ ≤ C | y 2 − y ′ 2 | ε 2 |≤ | x 2 − y 2 | � K 2 ( x 2 , y 2 ) − K 2 ( x 2 , y ′ if | y 2 − y ′ , | x 2 − y 2 | m + ε 2 2 , y 2 ) � CZ ≤ C | x 2 − x ′ 2 | ε 2 |≤ | x 2 − y 2 | � K 2 ( x 2 , y 2 ) − K 2 ( x ′ if | x 2 − x ′ . | x 2 − y 2 | m + ε 2 H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

Journ´ e’s class (cont.) � K 1 ( x 1 , y 1 ) � CZ ≤ C | x 1 − y 1 | n means that K 1 ( x 1 , y 1 ) is boubded on L 2 ( R m ) and its associated kernel K 1 ( x 1 , y 1 )( x 2 , y 2 ) = K ( x 1 , x 2 , y 1 , y 2 ) satisfies | K 1 ( x 1 , y 1 )( x 2 , y 2 ) | ≤ � K 1 ( x 1 , y 1 ) � CZ 1 | x 2 − y 2 | m , | K 1 ( x 1 , y 1 )( x 2 , y 2 ) − K 1 ( x 1 , y 1 )( x 2 , y ′ 2 ) | | y 2 − y ′ 2 | ε 2 |≤ | x 2 − y 2 | ≤ � K 1 ( x 1 , y 1 ) � CZ if | y 2 − y ′ , | x 2 − y 2 | m 2 | K 1 ( x 1 , y 1 )( x 2 , y 2 ) − K 1 ( x 1 , y 1 )( x ′ 2 , y 2 ) | | x 2 − x ′ 2 | ε 2 |≤ | x 2 − y 2 | ≤ � K 1 ( x 1 , y 1 ) � CZ if | x 2 − x ′ , | x 2 − y 2 | m 2 H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

Let � � � R n ψ ( y ) y α dy = 0 C ∞ ψ ∈ C ∞ 0 , 0 ( R n ) = c ( R n ) : for 0 ≤ | α | ≤ N p , n . Let n 1 = n , n 2 = m , ψ i ∈ C ∞ 0 , 0 ( R n i ) supported in the unit ball of R n i , and ψ i satisfy condition � ∞ � � � 2 dt i � � ψ i ( t i ξ i ) = 1 for all ξ i � = 0 , i = 1 , 2 . t i 0 For t i > 0 and ( x 1 , x 2 ) ∈ R n × R m , set ψ i t i ( x i ) = t − n i ψ ( x i / t i ) and i ψ t 1 t 2 ( x 1 , x 2 ) = ψ 1 t 1 ( x 1 ) ψ 2 t 2 ( x 2 ). H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

Product Hardy spaces H p ( R n × R m ) The product Littlewood-Paley g -function of f ∈ S ′ ( R n + m ) is defined by � � ∞ � ∞ � 1 / 2 � � � 2 dt 1 dt 2 � ψ t 1 , t 2 ∗ f ( x 1 , x 2 ) g ( f )( x 1 , x 2 ) = . t 1 t 2 0 0 For 0 < p ≤ 1, the product Hardy space is defined by � � H p ( R n × R m ) = f ∈ S ′ ( R n + m ) : g ( f ) ∈ L p ( R n + m ) with � f � H p ( R n × R m ) := � g ( f ) � L p ( R n + m ) . H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

The multiparameter product Hardy spaces H p ( R n × R m ) are much more complicated than the classical one-parameter Hardy spaces H p ( R n ). It was conjectured that Conjecture The product H 1 ( R × R ) could be characterized by rectangle atoms; that is, � � H 1 ( R × R ) = f : f = λ k a k , where a k are rectangle atoms � � and | λ k | < ∞ . H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

Here a rectangle atom is a function a ( x , y ) supported on a rectangle R = I × J satisfying � � � a � 2 ≤ | R | 1 / 2 and a ( x , y ) dx = a ( x , y ) dy = 0 . I J Carleson disproved this conjecture by constructing a counter-example of a measure satisfying the product form of the Carleson measure which is bounded on � f : f = � λ k a k , a k rectangle atoms and � � k | λ k | < ∞ , but is not bounded on H 1 ( R × R ). H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

Atomic decomposition for H p ( R n × R m ) In 1980, Chang and R. Fefferman [Ann. of Math.] proved � f ∈ H p ( R n × R m ) ⇐ ⇒ f = λ k a k , where � | λ k | p < ∞ and a k ( x , y ) are atoms; that is, 1 a k is supported in an open set Ω in R n × R m with | Ω | < ∞ ; 2 � a k � 2 ≤ | Ω | 1 / 2 − 1 / p ; 3 a k ( x , y ) = � R ∈M (Ω) a R ( x , y ), each “pre-atom” a R satisfies: • supp( a R ) ⊂ 4 R , R := I × J ( I a dyadic cube in R n , J a dyadic cube in R m ); • M (Ω) is the collection of all maximal dyadic rectangles � � � 1 / 2 ≤ | Ω | 1 / 2 − 1 / p ; R ∈M (Ω) � a R � 2 contained in Ω, and 2 � � a R ( x , y ) x α dx = a R ( x , y ) y β dy = 0 for 0 ≤ | α | ≤ N p , n and • 0 ≤ | β | ≤ N p , m . H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

H p ( R n × R m ) − L p ( R n + m ) boundedness R. Fefferman [Proc. Natl. Acad. Sci. USA (1986)] proved the following remarkable result. Theorem Let 0 < p ≤ 1 and T be a bounded linear operator on L 2 ( R n + m ). Suppose that there exist constants C > 0 and δ > 0 such that, for any H p ( R n × R m ) rectangle atom a supported on R , � ( γ R ) c | Ta ( x , y ) | p dxdy ≤ C γ − δ ∀ γ ≥ 2 . Then T extends to a bounded operator from H p ( R n × R m ) to L p ( R n + m ). H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee

H p ( R n × R m ) − L p ( R n + m ) boundedness (cont.) Here a rectangle atom a is a function supported on a rectangle R = I × J ( I a cube in R n , J a cube in R m ) such that 1 � a � 2 ≤ | R | 1 / 2 − 1 / p ; 2 � � I a R ( x , y ) x α dx = J a R ( x , y ) y β dy = 0 for 0 ≤ | α | ≤ N p , n and 0 ≤ | β | ≤ N p , m . R. Fefferman further proved that product singular integrals in Journ´ e’s class satisfy the above estimate and hence these operators are bounded from H p ( R n × R m ) to L p ( R n + m ). The boundedness of these operators on H p ( R n × R m ) is still open. H p ( R n × R m ) boundedness of CZO’s Ming-Yi Lee